1. Quadrilaterals and Coordinates Proof
Coordinate Proof for Parallelograms
Opening routine
Determine if the following quadrilateral is a parallelogram for the given values of the variables.
x = 6
y = 3.5

2. Topic IV: Quadrilaterals and Coordinate Proof

3. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Objective: Use coordinate geometry to prove
that a quadrilateral is parallelogram.
Essential Question: What criteria can be used in a coordinate proof to determine of a quadrilateral is a parallelogram?

4. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Vocabulary
Parallelogram: Is a quadrilateral with two pairs of parallel sides.
Rectangle: Is a quadrilateral with four right angles.
Rhombus: Is a simple quadrilateral whose four sides all have the same length.
Square: A square is a regular quadrilateral, which means that it has four equal sides and four equal angles.

5. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Vocabulary
Trapezoid: Is a quadrilateral that has a pair of opposite sides parallel. The sides that are parallel are called “bases”.
Isosceles trapezoid: Is a special type of trapezoid in which non-parallel sides and base angles are equal.
Kite: Is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

6. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms

7. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms

8. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms

9. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms

10. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms

11. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
To prove if a quadrilateral in the coordinate plane is a parallelogram, it can be used four different criteria:
Distance formula to determine if both opposite sides are congruent.
Slope formula to determine if both opposite sides are parallel.
Midpoint formula to determine if the diagonals bisect each other.
Distance formula and slope formula to determine if one pair of opposite sides is both parallel and congruent.

12. Quadrilaterals and Coordinates Proof
Test for Parallelogram in the Coordinate Plane
Guided Practice – WE DO

13. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Guided Practice – WE DO
The vertices of quadrilateral
JOHN are J (3, 1), O (3, 3),
H (5, 7) and N (1, 5). Use
the coordinate geometry to
prove that quadrilateral
JOHN is a parallelogram.

14. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Guided Practice – WE DO

15. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Guided Practice – WE DO
We will use the distance formula
and the slope formula to determine
if the quadrilateral is a parallelogram.

16. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Guided Practice – WE DO
Determine whether the figure with the
given vertices is a parallelogram,
P (5,3), Q (1,5), R (6,1), S (2,7)

17. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Guided Practice – WE DO
Determine whether the figure with the
given vertices is a parallelogram,
P (5,3), Q (1,5), R (6,1), S (2,7)
Then PQRS is a parallelogram

18. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Independent Practice - YOU DO
Worksheet “Coordinates Proof for Parallelogram”
Exercises from 1 to 4

19. Quadrilaterals and Coordinates Proof
Coordinates Proof for Parallelograms
Closure
Essential Question: What criteria can be used in a coordinate proof to determine of a quadrilateral is a parallelogram?